The breakage of large brittle particles by applying different material parameters has been modeled with the discrete
element method (DEM). The model accounts for incremental breakage where particles can get damaged without being
fragmented. Particles might fragment depending on their damage history, size, material strength and impact energy
involved. Repetitive breakup of progeny particles and simultaneous breakage of many particles at the same moment
is taken into account. The desire and requirement for modeling a changing particle population in multiphase flows is
very high, as many more observed phenomena in practice can be studied similarly. The size distribution of the progeny
is based on one single breakage index parameter t10, able to allow breakage into very few similar-sized fragments
towards an attrition-like progeny where one big fragment is produced next to several smaller ones. The velocity of
the fragments is derived from the momentum and energy conservation. Three examples have been chosen for further
discussion by highlighting the model’s strength and limitations. Grinding of particles inside a semi-autogenous mill
and missile fragmentation are used to analyze the time and space evolution of the main variables and for comparisons
with other models. Breakage of char particles inside a fluidized bed is discussed as a typical example of a multiphase
flow application.

General Note:

The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows

The breakage of large brittle particles by applying different material parameters has been modeled with the discrete
element method (DEM). The model accounts for incremental breakage where particles can get damaged without being
fragmented. Particles might fragment depending on their damage history, size, material strength and impact energy
involved. Repetitive breakup of progeny particles and simultaneous breakage of many particles at the same moment
is taken into account. The desire and requirement for modeling a changing particle population in multiphase flows is
very high, as many more observed phenomena in practice can be studied similarly. The size distribution of the progeny
is based on one single breakage index parameter ti0, able to allow breakage into very few similar-sized fragments
towards an attrition-like progeny where one big fragment is produced next to several smaller ones. The velocity of
the fragments is derived from the momentum and energy conservation. Three examples have been chosen for further
discussion by highlighting the model's strength and limitations. Grinding of particles inside a semi-autogenous mill
and missile fragmentation are used to analyze the time and space evolution of the main variables and for comparisons
with other models. Breakage of char particles inside a fluidized bed is discussed as a typical example of a multiphase
flow application.

Irreversible size reduction of solid material is often de-
sired or undesired in a large variety of applications.
Comminution and grinding should lead to a much
smaller particle population with a minimum energy sup-
plied to the system. Vice-versa in combustion fine char
particles are formed during fragmentation following the
flue gas before bur-out leading to a reduction in com-
bustion efficiency. Hence, a good understanding and de-
tailed study of fragmentation in general is important to
control these processes.
The majority of fragmentation models can be di-
vided into two groups. The first group uses empiri-
cal correlations where the fragmentation event is non-
discrete, which predict fragmentation in average. Most
models are very simple in nature and might deliver
quick trends in similar applications under similar cir-
cumstances. Therefore, these models may fall behind in
their reliability as fitted parameters may not cope with
e.g. non-homogeneous properties (materials, applied
forces, different flow pattern, etc). Inadequate informa-
tion about the fragment number and velocity, their con-
sequence and behavior of the remaining particle phase,
information about the particle history (e.g. how much it
got damaged before) and much more remain in the dark,
even for idealized properties.
The second group of fragmentation models consider
one single or very few discrete particles which might be
involved in a breakage event. DEM models are capa-
ble of modeling particle agglomerates representing one
global particle by many smaller ones. When it breaks, it
falls into fragment sizes depending on previously spec-
ified subparticles and if not their discretization remains
in vain. For these models the computational cost would
roughly increase by a factor of 103-104 (Cleary 2001)
compared to models without agglomerate subparticles.
The finite emelent method (or combined with the DEM
agglomerate approach) might deliver accurate predic-
tions of crack propagation and disruption inside parti-
cles. However, they are not tailored for collisional frag-
mentation where many particles might be involved.
Cleary (2001) was the first who developed a frag-
mentation model in DEM where numerous discrete sin-
gle particles can be fragmented and replaced by their
progeny. This method might be addressed to a third
group of fragmentation models which combine accu-
racy and efficiency which might be used for applica-
tions involving numerous particles as it can be found
in mills, crushers, fluidized beds etc. Cleary (2001)
stated, that the actual rules used in his code are still crude
and progress beyond fragmentation involving high speed

balls in cataracting streams is desired. No further de-
tailed description of his model has been published lately.
This paper presents a detailed description of a dis-
crete fragmentation model which might be categorized
into the same third group. The onset of fragmentation
is modeled by using a breakage probability which con-
siders incremental impact breakage by summation of ac-
cumulated damage. In principle, this model can be pro-
vided with any particle size distribution (PSD) how-
ever, a breakage index tl0 a single value to determine
the entire PSD has been proposed. This approach offers
several advantages over others as it depends on material
parameters only, it is approved to be valid for multiple
impact breakage and has been found valid for many brit-
tle materials. Discrete fragments are created depending
on the given PSD and packed randomly into their parent
particle volume with a minimum particle overlap. Every
fragment is assigned with a kinetic (velocity) component
derived from the momentum conservation and an elastic
(spring force) component derived from the energy equa-
tion. The model outcome has been compared to other
model outcomes with little deviation and further setting
parameters have been tested on a semi-autogenous mill
model according to our expectations. This fragmenta-
tion model has been developed in the CFD code called
MultiFlow coupled inside its soft-sphere DEM module
based on non-linear collision forces.

Fragmentation model

Most brittle materials can get damaged without being
fragmented. In this work, this phenomenon is referred
as incremental breakage (damage) initiated when the im-
pact energy exceeds a threshold energy Eo required for
extending flaws inside the material. The particle damage
is set equivalent to the probability of breakage which in
turn is mathematically expressed by a probability func-
tion introduced by Vogel and Peukert (2004) and modi-
fied for DEM applications by Morrison et al. (2007) ac-
cording to:

P = 1 exp -fMat x -(E Eo) ,

where Eo is the mass specific threshold energy which
a particle can absorb without fracture, E, is the mass
specific impact energy during the i'th impact, fMat is a
material parameter characterizing the resistance of par-
ticulate matter against fracture in impact comminution
and x is the particle size. Whenever the destroyed frac-
tion of the particle or the probability of breakage P is
high enough, body breakage (cleavage or shattering de-
pending on the impact energy) will occur (Tavares and
de Carvalho 2009).

Figure 1: (a) A brittle particle moves towards a wall, (b) when the maximum elastic energy is reached (see overlap
with the wall) and it comes to fragmentation, the parent particle is replaced by child particles which do not touch

the wall (c) the arrow indicates the direction and particles
pictures the progeny after fragmentation

The breakage index t10 indicates the cumulative mass
passing 1/10 of the parent particle diameter and is re-
lated to other t, values for many materials (Narayanan
and Whiten 1988). M is the maximum achievable tio
in a single breakage event and is related to other mea-
sures of rock strength (Bearman et al. 1997). Material
values for fMat and E0 can be found for rock in Bear-
man et al. (1997) or polymers, glass and limestone in
Vogel L. and Peukert W. (2005). For unknown mate-
rials, Napier-Munn et al. (1996, Chap.5) provide a de-
tailed description of how these data are measured. The
material parameter fMat is inversely dependent on the
well known fracture toughness.
This approach uses a one-parameter family of curves
to receive the progeny size distribution (Shi and Ko-
jovic 2007) and has been found valid for many brit-
tle materials under considering multiple impact break-
age. However, the progeny PSD can be replaced by any
other PSD function available (Rosin-Rammler, Gaudin-
Schumann,...) in case the tio approach does not cover
a certain material but it has the advantage of being in-
dependent of fitted data and machine specific testing
methods. Once the PSD is known, discrete fragments
of different diameters are created based on the fragment
size and remaining cumulative mass percent given by the
PSD curve. A minimum diameter di, need to be speci-
fied as a model setting parameter to avoid numerous tiny
fragments. Therefore, the overall number of fragments

are colored by their velocity (black=slow, white=fast) (d)

depend mainly on dm and M but also on the breakage
probability P obtained beforehand. Due to its complex-
ity, fragments are randomly placed inside the parent par-
ticle volume where fragments are inserted without other
parent particle or wall overlaps as can be seen in Fig. lb.
Their distance to each other is maximized to reduce the
overlapping child-child particle volume and to achieve
maximum code stability.
During fragmentation, kinetic energy of the parent
particle is transformed as fragments are halted by con-
tact with the surroundings, destroying fragment mo-
mentum and generating forces on the surroundings (Chi-
rone et al. 1982). This phenomenon is dependent on the
material and is in the present model considered by a mo-
mentum factor eMF. This factor can be interpreted as a
coefficient of restitution e for fragmentation which con-
siders that momentum goes into the formation of crack
extensions. In contrast, the total energy required for
fragmentation is often more than 100 times larger than
the energy required to produce new surface and which fi-
nally might get dissipated. Stretching and disruption of
intermolecular force fields require work be done where
almost all of this is recovered as kinetic energy when
the force fields separate and return to their unstressed
states (Bergstrom 1963). For that reason, the present
model does not consider a dissipative term in the en-
ergy equation but can be considered for future studies
as long as a reliable theory is provided. In this model,
the fragment velocity eq. (5) is computed from a general
definition of etot eq. (3) and the momentum equation (4)
by considering the momentum factor eMF as:

Vati(3)
etot = eMFe (3)
Vbi

Vbim + F dt M

tlo = M 1

-m vai (4)

Vai etot [V+ z dtj, (5)

where Vb, is the velocity of the parent particle before
impact, m and M are the mass of a fragment and the
parent particle, respectively. Eqs. (3-5) are used at the
moment of breakage only, where the sign prior to the
velocity during impact V' is always the same as Va, be-
cause the force to trigger fragmentation has passed its
culmination. Furthermore, these equations are valid in
the direction of the impact normal only. This integral
over the remaining time of the collision is approximated
by linearization and obeys eqs. (6-8).

F dt= FLt + 2t +... (6)

F2 = kn 0/2 i (7)

"F2

M
'2 =i -AS 1i [ F2 At2 +vAt (8)

Here, 6 is the particle overlap distance, At is the par-
ticle time step used, Fi is the current force associated
with the overlap during the fragmentation event, k, is a
spring constant in the normal impact direction, 4 is the
normal vector and v is the velocity of the colliding par-
ticle starting from zero. The integral is solved, when the
remaining 6, from eq. (8) becomes zero. It should be
pointed out, that each fragments momentum is a mass
weighted portion of the overall momentum leading to a
minimum kinetic energy of the fragments. At this stage,
the velocity of each fragment is the same but will change
when elastic forces on each fragment are considered (ar-
tifical overlap due to fragmentation).
Collisions between child particles are considered as
internal forces which do not change the total linear mo-
mentum of a system. Child particle collisions with the
wall or other parent particles (external forces) do not
exist (Fig. lb) at the moment of fragmentation as all
fragments are inserted without external overlap. Con-
servation of angular momentum is not considered in the
present model, as fragments might experience high shear
forces between them.
Artificial overlaps between child particles need to be
corrected in terms of their associated elastic energy. The
artificial overlap between child particles at the moment of
fragmentation is remembered in 6,em and a dimension-
less collision factor CF is applied to correct the associ-
ated artificial elastic energy. At each particle time step,
r6em is updated according to ,em = MIN(rem, 6)
as long as the collision is found in the collision list (the
collision exists). Eq. (11) is introduced to obtain CF by

The asterisk indicates geometrically correct but not ener-
getically correct values, the dash indicates energetically
correct but geometrically incorrect values after replace-
ment (AR) and j is a counter variable for the fragments.
For CF = 1, eq. (10) is the integration of eq. (7) and
corresponds to the elastic energy for the Hertzian contact
theory. The elastic energy stored by the particle until the
instance of fracture (second term in eq. (9)) is the well
known particle fracture energy (Baumgardt et al. 1975).
The CF value is the same for all fragment collisions cre-
ated by the same broken parent particle and acts within
6 < Srem only.

The present model relies on a few random numbers
for instance to convert the breakage probability into a
clear Dirac answer. Fragmentation is a very complex
phenomenon and experimental results are hardly repro-
ducible when it comes to exact measures. Therefore, an

Figure 3: (a) Forces acting between a wall and a recoiling ball at normal incidence for different Young's moduli and
(b) the breakage behavior inside the SAG-mill for the same particle stiffness

application is required which involves numerous parti-
cles to judge about the average onset over all breaking
particles. In this study a semi-autogenous mill has been
chosen to test different setting parameter and their effect
on the breakage frequency (onset).
Semi-autogenous mills are loaded with large heavy
balls and small charge particles which suppose to be
crushed by the balls. The geometric mill data are taken
from Djordjevic et al. (2006), with an initial mill diam-
eter of 1.19m and a length of 0.31m fitted with 14 metal
lifters each 40 mm in hight. The rotating speed is 70 %
of its critical speed (3.14 rad/s). For each simulation, 24
large balls (p C5001,,n,/lm3 and d 0.1m) and 714
charge particles (p = 2'50,', r/n3 and d 0.05m) are
loaded and grinded in batch mode. Industrial to lab-scale
applications often produce millions of particles down to
sizes of microns. DEM models cannot solve such prob-
lems within a reasonable computational time so that sim-
plifications are required herein, only the large parti-
cle fraction is considered. The discrete-fragmentation
model has been tested using self-specified particle prop-
erties to demonstrate their impact on the breakage fre-
quency. Therefore, charge particles are grouped into dif-
ferent bin sizes named as M1, M2,... which have been
kept within the /2 sequence. In this study, particularly
the mass reduction of particles in the top size (original
size) of d 0.05m (M1) is of interest as other size
classes depend simultaneously on a created and reduced
fraction.
Soft-sphere DEM models in general and their results
rely on the particle stiffness (Youngs modulus and Pois-
son ratio) and so does the present discrete fragmenta-
tion model. Fig. 3a shows the force acting between a

Table 1: Particle property settings

Variable

fMat,ref
(:X Eo)ref
Mref
d,.i.
eMF

Ewall,ref
Ep,ref

dball
dcharge,int

Value Units

0.9
0.15
10
0.0125
0.5
0.97
0.1
10+8
10+7
0.25
0.1
0.05

kg/Jm
Jm/kg

m

Pa
Pa

m
m

rebouncing ball hitting a wall at normal incidence in-
dicating that for a softer material the contact will last
longer with a smaller force magnitude. The same stiff-
ness values have been used for the particles grinded in-
side a semi-autogenous mill as depicted in Fig. 3b show-
ing that the softer the material the higher the breakage
frequency (for a low E0 value). This is because the time-
scale to allow fragmentation is much longer for smaller
(x Eo) values. All settings for the fragmentation model
which have not been modified are summarized in Table
1 except otherwise stated.
Next to the particle stiffness, the material parameter
fMat, the mass specific threshold energy E0 and the
particle size x do influence the onset of fragmentation
according to eq. (1). Vogel and Peukert (2004) indi-
cated that the product :x Eo is constant for all particle

Figure 4: (a) Grinding times for different values of x Eo (threshold energy to achieve damage) and (b) different fmat
values and their influence on the breakage frequency / grinding time and cumulative mass fractions of charge particles
in other bins

sizes, also used in the present model as a single setting
parameter so that its influence as a whole is tested and
shown in Fig. 4a. As a rough estimate, its dependency
can be assumed to be linear with the required grinding
time. The higher the threshold for damaging particles
the lower the breakage frequency. The influence of the
material parameters fMat is depicted in Fig. 4b. As in-
dicated earlier, fMat is inversely dependent on the well
known fracture toughness Kic, so that smaller values
for fMat causing longer grinding times.
Fig. 2 shows the particles inside the semi-autogenous
mill after 4.5 minutes of operation. The big blue balls
are not damaged at all while smaller particles from the
charge are colored according to their damage history.
Particles from the smallest bin size M5 are not allowed
to break further as they are restricted by d,,,, so that
their damage might reach 100%. This leads to misin-
terpretation and need to be changed in future studies.
The advantage of modeling the comminution in mills is
a systematic analysis of most fragmentation setting pa-
rameters available and to judge about their influence on
breakage in general. However, DEM limitations restrict
fragmentation modeling in terms of the applied particle
number, size and stiffness, so that particular industrial
applications have to be simplified when modeled.

Missile fragmentation

Fragment velocities indicate how the impact energy is
partitioned and how the fragment cloud expands into the
local surrounding. The present model cannot be applied
for crater formation studies and cannot handle melting

metallic projectiles during hypervelocity impact. Its
strength lies in the multiphase coupling often required
for fragmentation studies and that the partitioning of the
impact energy is material specific, although appropriate
parameters are rather difficult to obtain. The momentum
factor CMF is responsible for the fragment cloud forma-
tion in the present model. The more initial momentum
is required for crack extentions, the bigger is the frag-
ment cloud (the distance between fragments long after
impact), which corresponds to the white particles (Fig-
ure 5a), a low CMF and vice versa.

The present discrete-fragmentation model considers
particle kinetic and elastic energy separately like all soft-
sphere discrete element models. At the moment of frag-
mentation, the model solves the momentum equation to
obtain the kinetic energy for each fragment (1.veloc-
ity component) and solves the energy equation to ap-
ply the remaining impact energy in form of elastic en-
ergy between the fragments. The elastic energy is con-
verted into kinetic energy, when all fragments have lost
their mutual contact (2.velocity component). As this ap-
proach is novel for fragmentation models, it has been
compared to the hypervelocity-fragment-cloud model
developed by Schafer (2006). Both models have been
simplified using fragments of only one single size and
a total number of 428 fragments. Schafer's 2D frag-
mentation model spatially locates all projectile frag-
ments on the circumference of a circle (broken circle in
Fig. 5b), where fragments are assumed to be uniformly
distributed on that circle. In both models, wall frag-
ments have been considered in the momentum and en-
ergy equation to account for the two-component veloc-

Figure 5: (a) Effect of the momentum factor on the generated fragment cloud after normal incidence at the wall; white
fragments for eMF = 0.05 and black fragments for eMF = 0.95 and (b) comparison of the resulting fragment cloud
between the present 3D and Schafer's 2D fragmentation model after 27ps

ity approach. Model parameters used are summarized in
Table 2.

Table 2: Parameters used for a comparison of both mod-
els (present 3D and Schafers 2D model)
Variable Value

All points in Fig. 5b belong to the discrete-
fragmentation model and their average distance from the
cloud center is plotted as the solid circle. It has been
found that the first velocity component, given by the ini-
tial velocity derived from the momentum equation, dif-
fers by 2%. This value has been determined as the ratio
of each model's predicted distance between cloud cen-
ter and impact point. The second velocity component,
differs by 5%, determined as the ratio of the averaged
cloud (circle) diameter of both models. The compari-
sion of these two models is limited to the momentum
and energy equation and their direct impact on the frag-
ment cloud only as their field of application differs sig-
nificantly. However, it can be concluded that the imple-
mented energy and momentum equation give fairly ac-
curate results for missile velocities of up to 6700m/s to
predict reasonable fragment velocities.

Char Fragmentation in Hot Fluidized Beds

There are three types of fragmentation, namely attrition
(flaking), body breakage and percolation (Syred et al.
2007). Inside hot fluidized beds, char particles undergo
mainly attrition as particles degradate progressively to-
wards the particle center but ash and lose char material
at the surface tear off from the main body. For the sake
of simplicity, most existing models ignore fragmentation
or treat fragmentation as a shrinking process, the most
state of the art model is presented by Di Blasi (1995).
The present discrete-fragmentation model captures body
breakage into very few similar sized fragments towards
an attrition-like fragment size distribution as can be seen
in Fig. 6 by using the proposed tio approach. As frag-
ments are created with a lower density and diameter,
Stokes numbers become small and tiny fragments follow
closely the flue gas towards filters, pipe walls and cause
undesired deposit and might block the system. Further-
more, the carbon loss through the system by char frag-
ments is of particular interest, as it is desired to reuse the
char (information about the char quality/quantity) and
results might contribute to a more complete species bal-
ance of the whole system.

Fig. 7 illustrates the freeboard and splash zone of a
fluidized bed the area where most fragmenation events
are expected. The fragmentation model is based on
the toughness of different materials (e.g. changing lin-
ear with the particle density) but does not yet depend
on thermal stresses and pressure inside particles due to
volatile release which will be the focus for future work.

V.
S -

^yu
Ww

0.06

0.04

0.02

-0.02

-0.04

o

o o oo o
o o
0 0 00 t W ..0s j0:
o o 0 0 o

'o o" o
0 0 0

0 o
l- o
,~ I I I ,

-U.U
0

i l u i a i i l

Figure 6: Fragments created with the breakage index
t10, body-breakage into very few fragments (here two),
or attrition-like breakage into many fragments (here
656)

*

'.
S ,* .' *-.

Figure 7: Splash zone of a fluidized bed with initially
monosized particles and fragments generated by break-
age

Conclusions

A new model has been developed and tested to account
for discrete, incremental, repetitive and simultaneous
fragmenation events, particularly suitable for multiphase
flow applications. Three different examples have been
selected to demonstrate the strength and limitations of
the present model. Comparison with other models show
excellent agreement and parametric studies have demon-
stated model prediction to our expectation. The model
is able to fragment particles into an infinite number of
progeny particles as far as DEM-limitations concern.
The code delivers much information about the fragmen-
tation event, for instance the fragment velocity and tra-
jectory from the moment of breakage, the degree of par-
ticle damage accumulated in the past, or PSD's for gen-
erating breakage rate curves to judge the performance

of different applications. Information provided by the
model can support engineers in designing and optimiz-
ing all kind of applications where fragmentation is in-
volved without minimal use of empirical parameters.